In the following exercise, you will be given the opportunity to recall what happens when you add two linear functions. Given: \( f(x)=2 x+3, g(x)=x+2 \), \( h(x)=x-5, m(x)=-3 x-1 \) 1. \( p(x)=f(x)+g(x) \) a. Write the equation of \( p(x) \). b. In what way is the slope of \( p(x) \) related to the slope of \( f(x) \) and the slope of \( g(x) \) ? (Include increasing or decreasing in your answer. Also identify which function is increasing faster.) c. In what way is the \( y \)-intercept of \( p(x) \) related to the \( y \)-intercepts of \( f(x) \) and of \( g(x) \) ? d. Is \( p(x) \) the equation of a line? Justify your answer.

Let's work through each part of the exercise step by step. --- ### **1. \( p(x) = f(x) + g(x) \)** #### **a. Write the equation of \( p(x) \).** To find \( p(x) \), add the equations of \( f(x) \) and \( g(x) \): \[ \begin{align*} p(x) &= f(x) + g(x) \\ &= (2x + 3) + (x + 2) \\ &= 2x + 3 + x + 2 \\ &= (2x + x) + (3 + 2) \\ &= 3x + 5 \end{align*} \] **Answer:** \( p(x) = 3x + 5 \) --- #### **b. In what way is the slope of \( p(x) \) related to the slope of \( f(x) \) and the slope of \( g(x) \)? (Include increasing or decreasing in your answer. Also identify which function is increasing faster.)** **Slopes of the Functions:** - \( f(x) = 2x + 3 \) → Slope = **2** - \( g(x) = x + 2 \)  → Slope = **1** - \( p(x) = 3x + 5 \) → Slope = **3** **Analysis:** - The slope of \( p(x) \) is **3**, which is the sum of the slopes of \( f(x) \) and \( g(x) \) (i.e., \( 2 + 1 = 3 \)). - Since the slope of \( p(x) \) is positive (**3**), the function \( p(x) \) is **increasing**. - Both \( f(x) \) and \( g(x) \) are increasing functions because their slopes are positive (**2** and **1**, respectively). - Among the two, \( f(x) \) has a steeper slope (**2**) compared to \( g(x) \) (**1**), meaning \( f(x) \) is **increasing faster** than \( g(x) \). **Answer:** The slope of \( p(x) \) is the sum of the slopes of \( f(x) \) and \( g(x) \). Since both \( f(x) \) (slope = 2) and \( g(x) \) (slope = 1) are increasing, their sum \( p(x) \) also has a positive slope of 3, making it increasing. Additionally, \( f(x) \) increases faster than \( g(x) \) due to its larger slope. --- #### **c. In what way is the \( y \)-intercept of \( p(x) \) related to the \( y \)-intercepts of \( f(x) \) and of \( g(x) \)?** **\( y \)-Intercepts of the Functions:** - \( f(x) = 2x + 3 \) → \( y \)-intercept = **3** - \( g(x) = x + 2 \)  → \( y \)-intercept = **2** - \( p(x) = 3x + 5 \) → \( y \)-intercept = **5** **Analysis:** - The \( y \)-intercept of \( p(x) \) is the sum of the \( y \)-intercepts of \( f(x) \) and \( g(x) \) (i.e., \( 3 + 2 = 5 \)). **Answer:** The \( y \)-intercept of \( p(x) \) is the sum of the \( y \)-intercepts of \( f(x) \) and \( g(x) \). Specifically, \( 3 + 2 = 5 \), so \( p(x) \) has a \( y \)-intercept of 5. --- #### **d. Is \( p(x) \) the equation of a line? Justify your answer.** **Definition of a Linear Function:** A linear function is any function that can be written in the form \( y = mx + b \), where: - \( m \) is the slope. - \( b \) is the \( y \)-intercept. **Analysis of \( p(x) \):** - \( p(x) = 3x + 5 \) is in the form \( y = mx + b \), where \( m = 3 \) and \( b = 5 \). - There are no exponents or products of \( x \) that would make the function nonlinear. **Answer:** Yes, \( p(x) \) is the equation of a line because it is in the form \( y = mx + b \) with a constant slope (3) and a constant \( y \)-intercept (5). ---